Differentiable Power-Flow Optimization

Muhammed Öz; Jasmin Hörter; Kaleb Phipps; Charlotte Debus; Achim Streit; Markus Götz

cs.AIMar 30, 2026

Differentiable Power-Flow Optimization

Authors: Muhammed Öz, Jasmin Hörter, Kaleb Phipps, Charlotte Debus, Achim Streit, Markus Götz

Abstract

With the rise of renewable energy sources and their high variability in generation, the management of power grids becomes increasingly complex and computationally demanding. Conventional AC-power-flow simulations, which use the Newton-Raphson (NR) method, suffer from poor scalability, making them impractical for emerging use cases such as joint transmission-distribution modeling and global grid analysis. At the same time, purely data-driven surrogate models lack physical guarantees and may violate fundamental constraints. In this work, we propose Differentiable Power-Flow (DPF), a reformulation of the AC power-flow problem as a differentiable simulation. DPF enables end-to-end gradient propagation from the physical power mismatches to the underlying simulation parameters, thereby allowing these parameters to be identified efficiently using gradient-based optimization. We demonstrate that DPF provides a scalable alternative to NR by leveraging GPU acceleration, sparse tensor representations, and batching capabilities available in modern machine-learning frameworks such as PyTorch. DPF is especially suited as a tool for time-series analyses due to its efficient reuse of previous solutions, for N-1 contingency-analyses due to its ability to process cases in batches, and as a screening tool by leveraging its speed and early stopping capability. The code is available in the authors' code repository.

Figures & tables

Figure 1: Schematic view of the components of the energy grid and the current applications of power flow calculations.

Direct methods
Name	Description	Usage
DC approx. ( Qi et al., 2012 )	Remove nonlinearities and solve linear system	Screening
HELM ( Trias, 2012 )	Embed power-flow equation into holomorphic function	Guaranteed solution
Iterative methods based on fix-point-iterations
GS ( Grainger, 1999 )	Rewrite power equation as a fix-point iteration and update buses sequentially	Educational
SA ( Giraldo et al., 2022 )	Rewrite power equation as fix-point iteration and update buses simultaneously	Educational

Table 1: Power-flow solution methods. Abbreviations: DC: DC-Approximation, HELM: Holomorphic Embedding Load Flow Method, GS: Gauß-Seidel, SA: Successive Approximation, NR: Newton-Raphson, FDLF: Fast Decoupled Load Flow, DPF: Differentiable Power-Flow, GNS: Graph Neural Solver, KCLNet: Kirchhoff Current Law Network.

Figure 2: The figure depicts the operational graph of the power balance equation. The calculated power

S_{calc}

(depending on the current voltage vector) and the actual power

S_{bus}

are formed and their active/reactive components are used in

y_{calc}

and

y

to create the loss function. The green colored variables, namely the voltage magnitude of PV-buses and the voltage angle of PV and PQ buses, are trainable and are used as the voltage solution after training.

		IEEE118	Case9241pegase
Optimizer		Adam
Optimizer parameters	lr ( $\eta$ )	$0.0034$	$0.0001$
Optimizer parameters	decay ( $\beta$ )	$(0.979,0.963)$
Scheduler		Reduce lr on plateau
Scheduler parameters	factor	$0.547$
	patience	$41$

Table 2: Hyperparameters for DPF on the small 118-bus and the large 9,241-bus grid.

		IEEE118 $t=0$	IEEE118 $t>0$
Optimizer		Adam
Optimizer parameters	lr ( $\eta$ )	$0.03564$	$0.00027$
Optimizer parameters	decay ( $\beta$ )	$(0.9802,0.9440)$	$(0.7847,0.6624)$
Scheduler		Step lr	Reduce lr on plateau
Scheduler parameters	step	$100$
	$\gamma$	$0.773$

Table 3: Hyperparameter settings for Time Series on IEEE118 with time step

t

Figure 3: Comparison of our Differentiable Simulation (blue), NR (red) and the DC-approximation (green) on CPU without data loading time. Left: the IEEE-118 grid, right: case9241pegase grid. In both cases our differentiable simulation is slower than NR but the solution quality lies between NR and DC.

Figure 4: Scaling behavior of DPF (braces show the number of iterations), NR and DC. While on CPU (left) the scaling seems similar between the different approaches, on GPU (right) the better scaling behavior becomes apparent. Yet for a grid size of 9,241 NR is still faster, as our gradient-based approach needs about 1,000 iterations in its base form for convergence.

Figure 5: Solution distance of previous solutions (or initialization) to the solution of the next time step for the IEEE-118 grid. Subsequent time-steps have very similar grids, and as a result, very similar solutions.

Figure 6: Training curves for a time series application use case for the IEEE-118 grid. The first time step (left) needs a full fitting of roughly 1,000 iterations. Albeit slow, needs to be performed only once per grid and can possibly be replaced by Newton-Raphson. Subsequent time steps (right), using the first as initial guess, only require 100 iterations to reach a plateau in solution quality.

Figure 7: Normalized time per power-flow using batching on the case9241pegase grid. Without batching, a time of about

2\text{\,}\mathrm{ms}

per iteration and power-flow is needed while a time of

0.45\text{\,}\mathrm{ms}

per iteration and power-flow is achieved at a batch size of 64.

Figure 8: Scaling behavior with respect to grid characteristics. For node scaling (left), we increase the number of nodes in the grid by copying the case9241pegase grid and making few (

20*\#copies

) random connections throughout the combined grid. In case of edge scaling (right), we insert an increasing amount of random connections to the case9241pegase grid, adding two new non-zero entries (due to symmetry) per connection into the

Y_{bus}

matrix consisting of

37,655

non-zero entries. DPF is almost unaffected (rises from

4.7\text{\,}\mathrm{s}

5.2\text{\,}\mathrm{s}

) while the runtime of NR increases significantly.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Power-flow times using NR in $\mathrm{ms}$
Grid	Size	RC	No RC	TS	CA
case14	14	0.0204	0.0491	0.00790	0.0171
case118	118	0.1180	0.3206	0.0506	0.0746
case_illinois200	200	0.2447	0.5842	0.1032	0.1705
case300	300	0.4501	0.9606	0.2521	0.3341
case1354pegase	1,354	2.3303	4.0754	1.2324	1.5423

Appendix

Table 4: LightSim2Grid times on our device in

\mathrm{ms}

. Compared are times of power-flows for changing grids with and without recycling (RC), power-flows for fixed grids on time series (TS) and times for contingency analysis (TS). The runtime scaling appears to be between linear and quadratic. The fastest times are achieved for time series as the grid (and therefore the admittance matrix and the sparsity structure of the Jacobian) stays the same and can be reused.

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FMOPF: Latent Flow Matching with Constraint-Aware Interaction Priors for AC Optimal Power Flow

Zhilin Huang

China Southern Power Dispatching and Control Center, China Southern Power Grid

Constrained Generative Modeling Flow Matching Uncertainty Quantification Optimal Power Flow Constrained Optimization Power Systems

Open paper AskPDF arXiv

AC optimal power flow determines the minimum-cost generation dispatch under nonlinear power balance constraints and is solved thousands of times daily in electricity market operations. Learning a direct mapping from load conditions to OPF solutions can accelerate this computation, yet with deepening renewable penetration, a single optimal dispatch is no longer sufficient. Operators require a characterization of the distribution of feasible near-optimal solutions for risk quantification, sensitivity analysis, and multi-objective trade-off assessment. Supervised neural networks provide fast point predictions but cannot capture this conditional distribution. Diffusion-based generative models can sample diverse solutions in principle, yet existing methods operating in the raw state space exhibit degraded solution quality and fail to scale beyond medium-sized systems. We identify the root cause as the conflation of two distinct tasks within a single model. Compressing the high-dimensional OPF solution manifold is one task, and learning the conditional mapping from loads to that manifold is another. This paper presents FMOPF, a framework that resolves this conflation by decoupling compression from generation through latent flow matching and by explicitly modeling load-state coupling through a Constraint-Aware Interaction Prior Network. Experiments on four IEEE test systems demonstrate that FMOPF provides the most effective Newton-Raphson warm starts, achieves the lowest tail risk among generative methods, and is the first such method to scale to systems with several hundred buses while preserving full feasibility. Ablation studies confirm that the latent generation pipeline is a necessary condition for physical feasibility and that the interaction prior functions as a late-stage tail-risk controller.

Abstract

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